Finite Group Databasebeta

← #10767957

Subgroups — #10767957

Order 768.

1386 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86241C211yesmaximal, characteristic, complemented
2C3×#3109C211yesmaximal, characteristic, complemented
3#82347C211yesmaximal, characteristic, complemented
4#82310C211yesmaximal, characteristic, complemented
5#71812C211yesmaximal, characteristic, complemented
6#71168C214yesmaximal, complemented
7#70968C214yesmaximal, complemented
8#70832C211yesmaximal, characteristic, complemented
9#70827C211yesmaximal, characteristic, complemented
10#18178—31nomaximal, Sylow
11C2×#1009—21no—
12C3×#532C2211yescharacteristic
13C2×C3×C4×D4C2211yescharacteristic, complemented
14#5833—21no—
15C2×C4×#67—22no—
16#5726—21no—
17#5717—21no—
18#5676—22no—
19#5605C2214yes—
20D3×#121—22no—
21#5551—21no—
22#5550—22no—
23D4×D12—21no—
24#5477C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C3×#481C2212yescomplemented
27C3×#476—21no—
28C3×#459—21no—
29C3×#447C2212yescomplemented
30C3×#442—22no—
31C3×#432C2211yescharacteristic, complemented
32#5223C2212yescomplemented
33#5217C2212yescomplemented
34#5015—21no—
35#5014—22no—
36#5008—21no—
37#4993C2212yescomplemented
38#4968C2212yescomplemented
39#4960—21no—
40#4949—21no—
41#4932C2211yescharacteristic, complemented
42#4926C2211yescharacteristic, complemented
43#4537C2214yescomplemented
44#4470C2214yescomplemented
45#4468C2214yescomplemented
46#3496—31no—
47#3109—31no—
48#3109D311yescharacteristic, complemented
49#2397—31no—
50#1613—34no—
51#1595—34no—
52#1505—31no—
53#1505—31no—
54#1466—31no—
55C2×C3×#41—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C22×#67—22no—
59#1009—48no—
60C2×#239—22no—
61C2×D3×D4—22no—
62C2×D3×D4—24no—
63C2×D3×D4—22no—
64C2×#235—22no—
65C2×#235—21no—
66C22×D12—22no—
67C22×D12D411yescharacteristic
68C22×C4×D3D412yes—
69C2×C3×D8—22no—
70C3×C4⋊Q8D411yescharacteristic, complemented
71C3×#127D411yescharacteristic, complemented
72C3×#124—21no—
73C3×#121—21no—
74C3×#121C2312yes—
75C3×C4×D4—21no—
76C3×C4×D4—21no—
77C3×C4×D4—21no—
78C3×C4×D4—21no—
79C3×C4×D4—21no—
80C3×C4×D4D412yescomplemented
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C42C2311yescharacteristic
84C2×#217—22no—
85#946—22no—
86#939—24no—
87#938—22no—
88#938—22no—
89#937—24no—
90C2×#213—22no—
91#930—22no—
92#930—21no—
93#930C2312yes—
94C4×#67—48no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4C2312yes—
97C2×C4⋊Dic3D411yescharacteristic
98#924—22no—
99C2×Dic3⋊C4—22no—
100C2×C4×Dic3—21no—
101#895—22no—
102#894—24no—
103D3×C4⋊C4—22no—
104#884—24no—
105#883—24no—
106#882—24no—
107D3×C22⋊C4—24no—
108#875—21no—
109#874D411yescharacteristic, complemented
110C4×D12—21no—
111C4×D12—21no—
112C4×D12—21no—
113C4×D12—21no—
114C4×D12—21no—
115C4×D12D412yescomplemented
116#869D411yescharacteristic, complemented
117C3×#106—22no—
118C3×C4⋊C8—21no—
119C3×#102—22no—
120C3×#102—22no—
121C3×C22⋊C8C2312yes—
122C3×#97—21no—
123#832—22no—
124C3⋊#95—21no—
125C3⋊#95C2312yes—
126#830C2312yes—
127#809—22no—
128#808—22no—
129C4⋊(C3⋊C8)—21no—
130#803—21no—
131C2×H(5,𝔽2)—61no—
132#547—61no—
133#547—61no—
134#546—62no—
135#546—62no—
136D42—61no—
137D42—61no—
138#537—61no—
139#534—34no—
140#532—31no—
141#532C2×D311yescharacteristic
142C2×#121—62no—
143C2×C4×D4—31no—
144C2×C4×D4—62no—
145C2×C4×D4C2×D311yescharacteristic, complemented
146#481—32no—
147#481C2×D312yescomplemented
148#476—21no—
149#476—61no—
150#459—21no—
151#459—61no—
152#447—32no—
153#447C2×D312yescomplemented
154#442—22no—
155#442—62no—
156#432—31no—
157#432C2×D311yescharacteristic, complemented
158#399—32no—
159#399—61no—
160#392—32no—
161#392—61no—
162#376—31no—
163#327—34no—
164#327—34no—
165#323—34no—
166C24×C3—21no—
167C23×D3—24no—
168C23×D3C2×D412yes—
169C3×#41—44no—
170C2×C3×Q8—21no—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—42no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—44no—
181C22×C3×C4—41no—
182C22×C3×C4—41no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C2×#67—24no—
187C2×#67—22no—
188C2×#67—24no—
189C2×#67—44no—
190C2×#67—44no—
191C2×#67—44no—
192C22×Dic3—22no—
193C22×Dic3C2×D412yes—
194#239—48no—
195D3×D4—48no—
196D3×D4—48no—
197D3×D4—48no—
198D3×D4—84no—
199#235—48no—
200#235—44no—
201C2×D12—24no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—24no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—48no—
208C2×D12C2×D412yes—
209C2×C4×D3—24no—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—44no—
214C2×Dic6—21no—
215C3×D8—44no—
216C2×C3×C8—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4C2×D412yes—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—41no—
224C3×C22⋊C4—41no—
225C3×C42C2×D411yescharacteristic
226C3×C42C2×D411yescharacteristic
227C3×C42C2×D411yescharacteristic
228C3×C42C2×D411yescharacteristic
229#217—24no—
230#217—44no—
231#213—44no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—22no—
234(C2×D3)⋊C4—22no—
235(C2×D3)⋊C4—22no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—44no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3C2×D412yes—
242Dic3⋊C4—24no—
243Dic3⋊C4—44no—
244C4×Dic3—24no—
245C2×C3⋊C8—22no—
246H(5,𝔽2)—128no—
247C2×#41—21no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—61no—
251C22×D4—31no—
252C22×D4—62no—
253C22×D4—64no—
254C22×D4—62no—
255C22×D4—62no—
256C22×D4—62no—
257C22×D4C22×D311yescharacteristic
258C23×C4—21no—
259C23×C4—32no—
260C2×D8—22no—
261C2×D8—62no—
262C4⋊Q8—31no—
263C4⋊Q8#6711yescharacteristic, complemented
264#127—31no—
265#127#6711yescharacteristic, complemented
266#124—21no—
267#124—61no—
268#123—64no—
269#123—64no—
270C22⋊Q8—62no—
271C22⋊Q8—62no—
272#121—21no—
273#121—32no—
274#121—64no—
275#121—62no—
276#121—62no—
277#121—64no—
278#121—62no—
279#121—62no—
280#121—61no—
281#121C22×D312yes—
282#120—64no—
283#120—64no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—32no—
290C4×D4—61no—
291C4×D4—61no—
292C4×D4—61no—
293C4×D4—61no—
294C4×D4—61no—
295C4×D4—128no—
296C4×D4#6712yescomplemented
297C2×C4⋊C4—31no—
298C2×C4⋊C4—62no—
299C2×C4⋊C4—62no—
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—32no—
303C2×C22⋊C4—64no—
304C2×C22⋊C4—62no—
305C2×C22⋊C4—62no—
306C2×C42—61no—
307C2×C42C22×D311yescharacteristic
308#106—22no—
309#106—62no—
310C4⋊C8—21no—
311C4⋊C8—61no—
312#102—22no—
313#102—22no—
314#102—62no—
315#102—62no—
316C22⋊C8—32no—
317C22⋊C8C22×D312yes—
318#97—21no—
319#97—61no—
320#95—32no—
321#95—61no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—42no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—42no—
328C23×C3C22×D411yescharacteristic
329C22×D3—22no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—48no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—84no—
336C22×D3—84no—
337C22×D3AGL(1,ℤ/8)14yes—
338C3×Q8—42no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—42no—
342C3×D4—42no—
343C3×D4—41no—
344C3×D4—41no—
345C3×D4—44no—
346C3×D4—42no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—41no—
351C2×C3×C4—44no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—82no—
355C2×C3×C4—82no—
356C2×C3×C4#12011yescharacteristic
357C2×C3×C4#12011yescharacteristic
358C2×C3×C4#12011yescharacteristic
359C2×C3×C4#12011yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362#67—48no—
363#67—416no—
364#67—44no—
365#67—84no—
366C2×Dic3—24no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372D12—24no—
373D12—24no—
374D12—42no—
375D12—42no—
376D12—48no—
377D12—42no—
378C4×D3—44no—
379C4×D3—48no—
380C4×D3—44no—
381C4×D3—82no—
382Dic6—42no—
383C3×C8—42no—
384C3⋊C8—42no—
385C24—21no—
386C24—32no—
387C24—64no—
388#41—44no—
389#41—124no—
390#41—128no—
391#41—128no—
392C2×Q8—21no—
393C2×Q8—61no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—32no—
400C2×D4—42no—
401C2×D4—64no—
402C2×D4—64no—
403C2×D4—64no—
404C2×D4—62no—
405C2×D4—64no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—128no—
411C2×D4—124no—
412C2×D4—128no—
413C2×D4—128no—
414C2×D4—128no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—244no—
418C2×D4C2×#6712yes—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—32no—
422C22×C4—44no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—64no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—124no—
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4C23×D311yescharacteristic
434D8—44no—
435D8—124no—
436C2×C8—22no—
437C2×C8—62no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—32no—
442C4⋊C4—64no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—124no—
447C4⋊C4C2×#6712yes—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—64no—
453C22⋊C4—64no—
454C22⋊C4—62no—
455C22⋊C4—62no—
456C22⋊C4—62no—
457C22⋊C4—62no—
458C22⋊C4—124no—
459C22⋊C4—124no—
460C42—64no—
461C42D3×D411yescharacteristic
462C42D3×D411yescharacteristic
463C42C2×#6711yescharacteristic
464C42C2×#6711yescharacteristic
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—44no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—82no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3C2×#12011yescharacteristic
476C22×C3C2×AGL(1,ℤ/8)12yes—
477C2×D3—24no—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—44no—
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—82no—
485C2×D3—168no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—42no—
492C3×C4—81no—
493C3×C4#45312yes—
494Dic3—44no—
495Dic3—42no—
496Dic3—42no—
497C23—22no—
498C23—22no—
499C23—34no—
500C23—42no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—62no—
505C23—62no—
506C23—62no—
507C23—122no—
508C23—122no—
509C23—128no—
510C23—244no—
511C23—244no—
512C23C2×D3×D411yescharacteristic
513Q8—42no—
514Q8—122no—
515D4—22no—
516D4—22no—
517D4—42no—
518D4—41no—
519D4—41no—
520D4—42no—
521D4—44no—
522D4—42no—
523D4—64no—
524D4—64no—
525D4—122no—
526D4—122no—
527D4—128no—
528D4—128no—
529D4—1216no—
530D4—124no—
531D4—122no—
532D4—244no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—64no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—124no—
546C2×C4—128no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—124no—
550C2×C4—122no—
551C2×C4—242no—
552C2×C4#93711yescharacteristic
553C2×C4#93711yescharacteristic
554C2×C4#93711yescharacteristic
555C2×C4#93711yescharacteristic
556C2×C4C2×D3×D411yescharacteristic
557C2×C4C22×#6711yescharacteristic
558C8—42no—
559C8—122no—
560C2×C3—22no—
561C2×C3—82no—
562C2×C3—82no—
563C2×C3#300311yescharacteristic
564C2×C3C2×#45311yescharacteristic
565C2×C3#302511yescharacteristic
566D3—44no—
567D3—82no—
568C22—21no—
569C22—21no—
570C22—44no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—64no—
576C22—82no—
577C22—81no—
578C22—81no—
579C22—124no—
580C22—122no—
581C22—122no—
582C22—122no—
583C22—122no—
584C22—122no—
585C22—242no—
586C22—488no—
587C22D3×AGL(1,ℤ/8)12yes—
588C22C2×#93711yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—42no—
595C4—81no—
596C4—124no—
597C4—122no—
598C4—122no—
599C4#519812yes—
600C3#1817811yescharacteristic, complemented, Sylow
601C2—22no—
602C2—82no—
603C2—82no—
604C2—124no—
605C2—242no—
606C2#8293611yescharacteristic
607C2#8394411yescharacteristic
608C2C2×#519811yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3109C2
socleC22×C3C2×#120
hypercenter#3109D3
hypocenterC3#18178
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #86241C2semidirect
2. C3×#3109C2semidirect
3. #82347C2semidirect
4. #82310C2semidirect
5. #71812C2semidirect
6. #71168C2semidirect
7. #70968C2semidirect
8. #70832C2semidirect
9. #70827C2semidirect
13. C2×C3×C4×D4C22semidirect
26. C3×#481C22semidirect
29. C3×#447C22semidirect
31. C3×#432C22semidirect
32. #5223C22semidirect
33. #5217C22semidirect
37. #4993C22semidirect
38. #4968C22semidirect
41. #4932C22semidirect
42. #4926C22semidirect
43. #4537C22semidirect
44. #4470C22semidirect
45. #4468C22semidirect
48. #3109D3semidirect
70. C3×C4⋊Q8D4semidirect
71. C3×#127D4semidirect
80. C3×C4×D4D4semidirect
109. #874D4semidirect
115. C4×D12D4semidirect
116. #869D4semidirect
145. C2×C4×D4C2×D3semidirect
147. #481C2×D3semidirect
153. #447C2×D3semidirect
157. #432C2×D3semidirect
263. C4⋊Q8#67semidirect
265. #127#67semidirect
296. C4×D4#67semidirect
600. C3#18178semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#862412
C3×#31092
#823472
#823102
#718122
#711682
#709682
#708322
#708272
#181783